Foreman's preface gives a good account of the conventional view of
foundations of mathematics. However, the centrality of set theory is,
I think, not well-supported by the statement
"in essentially every area of mathematics with significant infinitary
content there are natural examples of statements independent of ZFC."
First of all, the "Absolute" part of mathematics, in which there are
very few independence-from-ZFC results, includes almost all of
mathematics as professionally practiced. Almost all mathematicians have
no compelling reason to care about higher set theory, until natural
independence-from-ZFC results are found in this part of mathematics
(Harvey Friedman's latest work continues his progress in this
direction, but there appears to be nothing in the 3-volume, 2197-page
Foreman-Kanamori handbook about Friedman's program.)
Secondly, most of the natural independence-from-ZFC statements in the
non-Absolute part of mathematics, such as the Continuum Hypothesis,
have no additional consistency strength and can be shown independent by
forcing arguments that can be formulated syntactically or in a much
smaller set-theoretic universe that does not essentially depend on "the
higher infinite".
This is not to say that there is anything wrong with the conventional
ZFC-foundation, which as Foreman points out has many practical
advantages. However the relevance of most of the contents of the
Foreman-Kanamori handbook to ordinary mathematics remains to be
established, and Foreman's statement quoted above seems far too strong.
-- JS
-----Original Message-----
From: Marc Alcobé <malcobe at gmail.com>
To: fom at cs.nyu.edu
Sent: Sat, Apr 24, 2010 12:36 pm
Subject: [FOM] Foreman's preface to HST
Dear FOMers,
In the spirit of the beginnings of this mailing list, please let me
ask you what are your opinions about Matthew Foreman's preface to the
(at least for me) long awaited Handbook of Set Theory (Foreman &
Kanamori eds.). There, Matt explains what he thinks is "a
well-accepted conventional view of foundations of mathematics".
I personally think that his view is a bit biased in favour of the kind
of research developed within standard set theory, as one would expect
in such a publication. But since I am no expert in the broad field of
the foundations of mathematics I feel compelled to learn about better
informed points of view.
I apologise for quoting the relevant statements in order to allow as
many people as possible to take part in the discussion.
"
[...] With roots in the Elements, the distinctive methodology of
mathematics has become proof. Inevitably two questions arise: What are
proofs? and What assumptions are proofs based on?
[...] Progress in the 19th and 20th centuries led to the understanding
of logics involving quantifiers as opposed to propositional logic and
to distinctions such as those between first and second-order logic.
With the semantics developed by Tarski and the compactness and
completeness theorems of Gödel, first-order logic has become widely
accepted as a well-understood, unproblematic answer to the question
'What is a proof?'
The desirable properties of first-order logic include:
• Proofs and propositions are easily and uncontroversially recognizable.
• There is an appealing semantics that gives a clear understanding of
the relationship between a mathematical structure and the formal
propositions that hold in it.
• It gives a satisfactory model of what mathematicians actually do:
the “rigorous” proofs given by humans seem to correspond exactly to
the “formal” proofs of first-order logic. Indeed formal proofs seem to
provide a normative ideal towards which controversial mathematical
claims are driven as part of their verification process.
While there are pockets of resistance to first-order logic, such as
constructivism and intuitionism on the one hand and other alternatives
such as second-order logic on the other, these seem to have been swept
aside, if simply for no other reason than their comparative lack of
mathematical fruitfulness.
To summarize, a well-accepted conventional view of foundations of
mathematics has evolved that can be caricatured as follows:
Mathematical Investigation = First-Order Logic + Assumptions
This formulation has the advantage that it segregates the difficulties
with the foundations of mathematics into discussions about the
underlying assumptions
rather than into issues about the nature of reasoning.
So what are the appropriate assumptions for mathematics? It would be
very desirable to find assumptions that:
1. involve a simple primitive notion that is easy to understand and
can be used to “build” or develop all standard mathematical objects,
2. are evident,
3. are complete in that they settle all mathematical questions,
4. can be easily recognized as part of a recursive schema.
Unfortunately Gödel’s incompleteness theorems make item 3 impossible.
[...] This inherent limitation is what has made the foundations of
mathematics a lively and controversial subject. Item 2 is also
difficult to satisfy. [...] The underlying primitive notions used to
develop standard mathematical objects are combined in very complicated
ways. The axioms describe the operations necessary for doing this and
the test of the axioms becomes how well they code higher level objects
as manipulated in ordinary mathematical language so that the results
agree with educated mathematicians’ sense of correctness.
Having been forced to give up 3 and perhaps 2, one is apparently left
with the alternatives:
2'. Find assumptions that are in accord with the intuitions of
mathematicians well versed in the appropriate subject matter.
3'. Find assumptions that describe mathematics to as large an extent
as is possible.
With regard to item 1, there are several choices that could work for
the primitive notion for developing mathematics, such as categories or
functions. With no a priori reason for choosing one over another, the
standard choice of sets (or set membership) as the basic notion is
largely pragmatic. Taking sets as the primitive, one can easily do the
traditional constructions that “build” or “code” the usual
mathematical entities [...] of the common objects of mathematical
study.
[...] ZFC [...] is pragmatic in spirit; it posits sufficient
mathematical strength to allow the development of standard
mathematics, while explicitly
rejecting the type of objects held responsible for the various
paradoxes,such as Russell’s.
[...] It is routine for normal axiomatizations that serve to synopsize
an abstract concept internal to mathematics to have independent
statements, but more troubling for axiom systems intended to give a
definitive description of mathematics itself. However, independence
phenomena are now known to arise from many
directions; in essentially every area of mathematics with significant
infinitary content there are natural examples of statements
independent of ZFC.
[...] Proposers of augmentations to ZFC carry the burden of marshaling
sufficient evidence to convince informed practitioners of the
reasonableness,
and perhaps truth, of the new assumptions as descriptions of the
mathematical universe. (Proposals for axiom systems intended to
replace ZFC carry
additional heavier burdens [...])
One natural way that this burden is discharged is by determining what
the supplementary axioms say; in other words by investigating the
consequences
of new axioms. This is a strictly mathematical venture. The theory is
assumed and theorems are proved in the ordinary mathematical manner.
Having an extensive development of the consequences of a proposed
axiom allows researchers to see the overall picture it paints of the
set-theoretic universe, to explore analogies and disanalogies with
conventional axioms, and judge its relative coherence with our
understanding of that universe.
[...] Many axioms or independent propositions are not related by
implication, but rather by relative consistency results[...]. A
remarkable meta-phenomenon has emerged. There appears to be a central
spine of axioms to which all independent propositions are comparable
in consistency strength. This spine is delineated by large cardinal
axioms. There are no known counterexamples to this behavior.
Thus a project initiated to understand the relationships between
disparate axiom systems seems to have resulted in an understanding of
most known natural axioms as somehow variations on a common theme—at
least as far as consistency strength is concerned. This type of
unifying deep structure is
taken as strong evidence that the axioms proposed reflect some
underlying reality and is often cited as a primary reason for
accepting the existence of
large cardinals.
The methodology for settling the independent statements, such as the
Continuum Hypothesis, by looking for evidence is far from the usual
deductive
paradigm for mathematics and goes against the rational grain of much
philosophical discussion of mathematics. This has directed the
attention of
some members of the philosophical community towards set theory and has
been grist for many discussions and message boards. However
interpreted,
the investigation itself is entirely mathematical and many of the most
skilled practitioners work entirely as mathematicians, unconcerned
about any philosophical anxieties their work produces.
Thus set theory finds itself at the confluence of the foundations of
mathematics, internal mathematical motivations and philosophical
speculation. Its
explosive growth in scope and mathematical sophistication is testimony
to its intellectual health and vitality. [...]
"
Thank you in advance.
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